The influence of different geometries of matrix/scaffold on the response of a bone and resorbable material mixture with voids

A 2-D dimensional sample made of natural bone tissue and artificial bioresorbable material is numerically investigated in order to study the influence of different geometries of the assemblage of matrix and scaffold. With the specific tools of the Mixture theory we consider the solid matrix with evolving apparent mass densities (rb for bone and rm for material) to describe bone tissue synthesis and resorption when a bio-resorbable material of the kind used in bone reconstruction is present (see e.g. [1, 2, 3]). To take porosity effects in account the adopted model is derived from the Nunziato-Cowin theory developed for porous solids in which the matrix material is linearly elastic and the interstices are void of material. In detail, to describe the mechanical phenomena which influence the porosity variation we introduce, following the aforementioned theory [4], an independent kinematic degree of freedom, namely the change in volume fraction from the reference volume fraction, namely x = (rb+rm)/rMax-(rb R+rmR)/rMax, with rMax the maximal density without pores. It is well established that exercise results in increased bone mass, while unloading due to immobilization, bedrest, and weightlessness results in bone atrophy. The strains induced by external loads are sensed by mechanoreceptors, primarily on osteocytes which essentially transduce the mechanical signals into biological signals. These biological signals are able to trigger bone remodeling by directing osteoblast activity and osteoclastic resorption. [1] T. Lekszycki and F. dell’Isola. A mixture model with evolving mass densities for describing synthesis and resorption phenomena in bones reconstructed with bio-resorbable materials. J. Applied Math and Mech. (ZAMM), 92:426–444, 2012. [2] A. Madeo, T. Lekszycki, and F. dell’Isola. A continuum model for the bio-mechanical interactions between living tissue and bio-resorbable graft after bone reconstructive surgery. Comptes Rendus Mécanique, 339:625–640, 2011. [3] A. Madeo, D. George, T. Lekszycki, M. Nierenberger, and Y. Rémond. A second gradient continuum model accounting for some effects of micro-structure on reconstructed bone remodeling. Comptes Rendus M´ecanique, 340:575–589, 2012. [4] S. C. Cowin and J. W. Nunziato. Linear elastic materials with voids. J. Elasticity, 13:125–147, 1983.